Suppose you meet a friend who you know for sure has two children, and he says: “I have a son born on Tuesday.” What is the probability that the second child of this man is also a son?

People argue about this problem a lot. Although I’ve discussed similar problems in the past, this particular problem has several interesting twists. See if you can identify them.

First, let us agree on some basic assumptions:

Sons and daughters are equally probable. This is not exactly true, but it is a reasonable approximation.

For our purposes, twins do not exist. Not only is the proportion of twins in the population small, but because they are born on the same day, twins might complicate the calculation.

All days of the week are equally probable birthdays. While this can’t actually be true — for example, assisted labors are unlikely to be scheduled for weekends — it is a reasonable approximation.

Now let us consider the first scenario. A father of two children is picked at random. He is instructed to choose a child by flipping a coin. Then he has to provide information about the chosen child in the following format: “I have a son/daughter born on Mon/Tues/Wed/Thurs/Fri/Sat/Sun.” If his statement is, “I have a son born on Tuesday,” what is the probability that the second child is also a son?

The probability that a father of two daughters will make such a statement is zero. The probability that a father of differently-gendered children will produce such a statement is 1/14. Indeed, with a probability of 1/2 the son is chosen over the daughter and with a probability of 1/7 Tuesday is the birthday.

The probability that a father of two sons will make this statement is 1/7. Among the fathers with two children, there are twice as many who have a son and a daughter than fathers who have two sons. Plugging these numbers into the formula for calculating the conditional probability will give us a probability of 1/2 for the second child to also be a son.

Now let us consider the second scenario. A father of two children is picked at random. If he has two daughters he is sent home and another one picked at random until a father is found who has at least one son. If he has one son, he is instructed to provide information on his son’s day of birth. If he has two sons, he has to choose one at random. His statement will be, “I have a son born on Mon/Tues/Wed/Thurs/Fri/Sat/Sun.” If his statement is, “I have a son born on Tuesday,” what is the probability that the second child is also a son?

The probability that a father of differently-gendered children will produce such a statement is 1/7. If he has two sons, the probability will likewise be 1/7. Among the fathers with two children, twice as many have a son and a daughter as have two sons. Plugging these numbers into the formula for calculating the conditional probability gives us a probability of 1/3 for the second child to also be a son.

Now let us consider the third scenario. A father of two children is picked at random. If he doesn’t have a son who is born on Tuesday, he is sent home and another is picked at random until one who has a son that was born on Tuesday is found. He is instructed to tell you, “I have a son born on Tuesday.” What is the probability that the second child is also a son?

The probability that a father of two daughters will have a son born on Tuesday is zero. The probability that a father of differently-gendered children will have a son who is born on Tuesday is 1/7. The probability that a father of two sons will have a son born on Tuesday is 13/49. Among the fathers with two children, twice as many have a son and a daughter than two sons. Plugging these numbers into the formula for calculating the conditional probability will give us a probability of 13/27 for the second child to also be a son.

Now let’s go back to the original problem. Suppose you meet your friend who you know has two children and he tells you, “I have a son born on Tuesday.” What is the probability that the second child is also a son?

What puzzles me is that I’ve never run into a similar problem about daughters or mothers. I’ve discussed this math problem about these probabilities with many people many times. But I keep stumbling upon men who passionately defend their wrong solution. When I dig into why their solution is wrong, it appears that they implicitly assume that if a man has a daughter and a son, he won’t bother talking about his daughter’s birthday at all.

I’ve seen this so often that I wonder if this is a mathematical mistake or a reflection of their bias.

How to solve the original problem? The problem is under-defined. The solution depends on the reason the father only mentions one child, or the Tuesday.

The funny part of this story is that I, Tanya Khovanova, have two children. And the following statement is true: “I have a son born on Tuesday.” What is the probability that my second child is a son?

24 Comments

Robert E. Harris:

Interesting dilemma. I really don’t think it’s indicative of bias, but there’s one way to test for sure: pose the question to two statistically significant groups of men, except, one of the groups gets asked the dual question, “You meet a friend who says ‘I have a daughter born on Tuesday’, what’s the probability his other child is also a daughter?” If it’s really a matter of misogynist bias, there should be a statistically significant qualitative difference in the reactions of the two groups of men. Interesting experiment, for certain…

Tipa:

colorblind:

What a coincidence. I also have two children. I have a son born on Tuesday (or maybe it’s Saturday, I’ll have to ask my wife…) and, whats more, my second child is a son.

His three sisters, the older one and the two younger, never let him forget he’s the only son.

As an only son myself (with two sisters), I remember being told by my parents part of the reason they had me was so I could carry on the family name. As the face of my family name, I would suggest we assume the father would automatically talk about the person who is going to reflect their family name for the future first. I.e. it’s an ego thing more than a bias. Not saying it’s right, but it is what it is.

Without using any information outside of this post, I would say that Tanya’s statement is an example of the third scenario. She is writing about a math problem with the statement “I have a son born on Tuesday” and notes “Hey, that happens to be true about me.” So the (mathematical) answer should be 13/27, as described above.

What do you think is more probable:
1. I started writing about Tuesday and then discovered that a have a son born on Tuesday.
2. I started writing about Wednesday (or about any other day), then checked that I have a son born on Tuesday and changed my essay to Tuesday.

Your second case had not occurred to me, as you described it as “the funny part of the story” and it wouldn’t be so funny if you had changed it. But now that you’ve raised the question, you’ve introduced some doubt, so let’s look at your second case.

Now, you could not have changed “daughter” to “son” since you state in the post that you have never heard a similar problem about daughters. Also, given Case 2, you do not have a son born on Tuesday, or you would not have needed to change it. There are two ways that you could have a boy and a girl, seven choices for the girl’s birth day and six choices for the boy’s birth day for a total of 84. There is one way to have two boys, and six choices for each son’s birthday for a total of 36. So the chances that your other son is a boy would be 36/(84+36) or 3/10.

So if Case 1 is true, then the probability is 13/27 that your other child is a son. If Case 2 is true, then the probability is 3/10.

To find which of the two cases is more probable would involve more psychology than math. How likely would it be for Tanya to change the problem, given that it does not apply to her as written, so that the question now does apply to her and she can say that it is funny. This is a matter of opinion, so I will define it as probability p. So in Case 1, she will write this post with probability 1 and in Case 2 she will write this post with probability p.

Given a person chosen randomly from the population of people with two children (and at least one son), how likely is it that this person will have a son born on Tuesday? If the parent has differently-gendered children, the probability is 1/7. If both children are boys, the probability is 13/49. The former being twice as likely as the latter, the total probability is 9/49.

So, a random math blogger writing about this problem has a 9/49 chance of having the problem apply to her (assuming she has two children including at least one son, which is given for you).

So the overall probability that your other child is a son, now that you have introduced doubt about the original problem, should be…

(9/49) * (13/27) + (40/49) * (3/10) = 1/3

… if the probability that you would change the problem and call it funny is 1. Given that we’ve defined it as p, I believe the solution is…

(9/(9+40p)) * (13/27) + ((40p)/(9+40p)) * 3/10

The range of solutions seems to be from 13/27 (where p=0) to 1/3 (where p=1). So the more likely you are to change the problem, the less likely it is that your other child is a son.

So another question might be: if we were to add the information that you have two sons, how likely is it that the original question said Tuesday?

Regarding the actual distribution of birthdays over a given week, this Stanford page gives us a neat chart which demonstrates visually the striking difference between weekend and weekday births. The chart deals with births within 1978.

Assuming it’s a Markov process, the probability
that the other child is a boy, hence that both
are boys, is .50 (Tuesday irrelevant as you’d
naively assume).

If you think otherwise, let me pose another riddle:
“I have two children. One is a >>girl<>girls<<?”

Now, you’ve got to come up with the same answer, right?
And you’ve got to make both these answers fit in with a
total probability, summed over all “states”, however
you’re counting them, of 1. So, show me how you’re
going to make that work sensibly with anything other
than 50-50 for both original riddles?

[…] me a variant of the two boys problem that has been generating a lot of discussion (here, here, here and here among other places). Appropriately enough, all of this discussion emerged from the most […]

[…] recently wrote two pieces about the puzzle relating to sons born on a Tuesday: A Son born on Tuesday and Sons and Tuesdays. I also posted a beautiful essay on the subject by Peter Winkler: Conditional […]

Steve:

“The probability that a father of two sons will have a son born on Tuesday is 13/49”
I thought this would be the figure if he said something along the lines of “I have _one_ son born on a Tuesday”, if he said “I have _a_ son born on Tuesday” would the probability not be just 14/49?

myles:

It is obvious that something is very wrong with any analysis that produces a probability result of anything other than .5 for the man having 2 sons. We know he has two kids and we know one of them is a boy. Probability is about what we don’t know. The only thing we don’t know is whether the remaining kid is a boy or girl. That’s 50/50. I’m not a mathematician but I’m surprised that some people, brighter than me, think otherwise. Let me ask them if they would be willing to place bets on the greater likelihood of the 2nd child being a girl. And how can they possibly explain that if the man had said that he has one son born on the 7th July of a non-leap year, then there is a different probability to him having a girl as his 2nd child than if he had said that he had a son born on a Tuesday. It’s just ridiculous! The whole universe would have a completely different logic system to the one we have now, and we would not be able to recognise it. Let me ask you, if you were an Intelligence Agency person, and you knew some terrorist in hiding had one son born on a Tuesday, and another child about whom nothing is known, would you go and report to your superiors that the odds are his having a daughter. You’d be straight out the door, and for good reason.

Carolyn Hope-Jones:

Re Boy v Girl

Traditionally ‘he’ can be either ‘he’ or ‘she’ in legal docs etc, as ‘man’ in the human sense includes ‘woman’. So we are slightly indoctrinated to talk about a boy rather than a girl. To mention a girl instead would seem to be a little feminist.

However, I feel the parent should be a woman rather than a man. Apart from the traditional role women have with children, it makes life easier when one knows that when one refers to ‘he’ one is talking about the child, and that ‘she’ refers to the parent.

But possibly most mathematicians are men (?) – I don’t know this (I’m not one) – so they would pose the problem from their viewpoint.

Carolyn Hope-Jones:

Re your final (personal) scenario, I agree with Bill that the answer is 13/27, and it’s very nice to find a realistic scenario where this answer occurs!

Of course, you might be lying (with good reason: it makes a good problem), but I think one has to discount this fact as the puzzle would not make sense. You certainly couldn’t take any bets on it.

However, I’m having trouble with your cat-amongst-the-pigeons post of 9 April (sorry – I’ve only seen this website today). I had thought this puzzle had originated elsewhere with the Tues spec and therefore you would be unlikely to have changed it to (WLOG) a Wed spec unless you actually had such a boy. I suppose you could then have changed it back again to Tues if you later discovered that your other boy was born on a Tues, in which case the chances of having 2 boys is 1.

But assuming for the moment that you did start arbitrarily with BoyW and then found at least one of your children was in fact a boy born on Tues, and so changed the puzzle to the BoyT spec, then I think the chances of you having 2 boys goes down from 13/27 to 11/25 as the BoyT/BoyNotT and BoyNotT/BoyT categories change to BoyT/BoyNotTorW and BoyNotTorW/BoyT – i.e. from 12 perms to 10 perms.

This is a different answer to Bill’s 3/10, and as he’s a mathematician and I’m not, he is likely to be right. But where is my mistake?

Carolyn Hope-Jones:

With regard to which is more probable (Tanya’s post of 9 April): 1. That you started writing about Tues and then discovered that you had a son born on Tues, or 2. You started writing about Wed then checked that you had a son born on Tues and changed your essay to Tues:-

This depends on whether you would change to Tues if you had a Tues boy, even if you also had a boy of the type you were first writing about (Wed).

If you were going to change to the Tues spec wherever possible, then I would have thought that Cases 1 and 2 are equally probable. This is if you are just considering Wed. If you mean all days other than Tues, then Case 2 is 6 times as likely as Case 1.

However, if you were only going to change to Tues if you did not have a Wed boy, then the chances of having a Wed boy being (1/7 + 1/7 + 13/49) divided by 3 = 9/49(?), the probability of changing to the Tues spec would be 40/49 instead of 1. If this type of change applies to all non-Tues days, then the chances of changing to Tues would be 40/49 x 6 = 240/49, i.e. about 5 times instead of 6. I don’t think there should be a reduction as I don’t think any double counting has gone on, but I’m not at all sure.

In conclusion, considering Wed only, Cases 1 and 2 are equally probable if you change to the Tues spec wherever possible, but Case 1 is more likely by about 1/5 (I don’t know how to work it out exactly!) if you only change when you don’t have a Boy Wed. And if you are considering all non-Tues days, then Case 2 is 6 times more probable than Case 1 in the ‘change whenever possible’ scenario, and about 5 times more probable in the latter scenario.

This is all assuming that you would DEFINITELY change if you had a Tues boy and didn’t have a boy of the day you were writing.

Carolyn Hope-Jones:

Oh dear, I must shut up – I seem to be talking to myself. But perhaps the post I said was nonsense, isn’t. I get confused with my scenarios. But the 11/25 result (for having 2 boys if you changed from Boy Wed to Boy Tues) perhaps is OK, assuming that if you were writing about Boy Wed and found you had a Boy Wed, you wouldn’t bother to change the spec to Boy Tues if your other child was a boy born on Tuesday.